Research Design in Occupational EducationCopyright 1997. James P. Key. Oklahoma State
University
Except for those materials which are supplied by different departments of the University
(ex. IRB, Thesis Handbook) and references used by permission.

MODULE S1

DESCRIPTIVE STATISTICS

All educators are involved in research and statistics to a degree. For
this reason all educators should have a practical understanding of research design. Even
if an educator is not involved in conducting research, he or she must be involved in the
interpretation of the findings of others to remain dynamic in their teaching.

The primary use of descriptive statistics is to describe information or
data through the use of numbers (create number pictures of the information). The
characteristics of groups of numbers representing information or data are called
descriptive statistics. Descriptive statistics are used to describe groups of numerical
data such as test scores, number or hours of instruction, or the number of students
enrolled in a particular course.

Descriptive statistics - Numbers which are used to describe
information or data or those techniques used to calculate those numbers.

Variable (x) - A measurable characteristic. Individual
measurements of a variable are called varieties, observations, or cases.

Population (X) - All subjects or objects possessing some common
specified characteristic. The population in a statistical investigation is arbitrarily
defined by naming its unique properties.

Parameter - A measurable characteristic of a population. A
measurable quantity derived from a population, such as population mean or standard
deviation.

Sample - A smaller group of subjects or objects selected from a
large group (population).

Statistic - A measure obtained from a sample. It is a measurable
quantity derived from a sample, such as the sample mean or standard deviation.

Frequency graph - A picture depicting the number of times an
event occurred.

Bar graph or histogram - A frequency graph with number of blocks
or length of bar representing the frequency of occurrence.

Frequency polygon - A modification of the bar graph with lines
connecting the midpoints of the highest point on each bar.

Frequency curve - A modification of a frequency polygon with the
sharp corners rounded. The area under the connecting line of the bar graph, frequency
polygon, and frequency curve are equivalent and represent frequency of occurrence.

Mean (µ) or () Arithmetical mean
- A number having an intermediate value between several other numbers in a group from
which it was derived and of which it expressed the average value. It is the simple average
formed by adding the numbers together and dividing by the number of numbers in the group

.

Median - The mid point in a set of ranked numbers.

Mode - The number which occurs most often in a group of numbers.

Range - The difference in the highest score and the lowest score
in a set of scores. The range is obtained by subtracting the low score from the high score
R = xh - xl.

Variance - The
mean of the squared deviations of individual numbers from the mean of the group of numbers
; the square of the standard deviation .

Standard deviation
- A measure of the deviation of individual numbers from the mean
of the group of numbers. It is the mean or average deviation of those numbers from the
mean of the set of numbers .

The use of symbols is a convenient way of expressing parameters or
statistics in research work. The following symbols are generally used to express
parameters and statistics.

Parameter

Statistic

Observation
of Variable

X

x

Sum

(Sigma)

Number
of observations

N

n

Mean

µ
(Mu)

Variance

(little sigma)

s2

Standard
Deviation

(little sigma)

s
or SD

Note that for the mean, variance, and standard deviation lower case
Greek letters are used to symbolize parameters, while lower case Roman letters are used to
symbolize statistics.

In the examples given, the raw scores for the 10 pt. quiz are:

10 9 8 8 7 7 6 6 5 4 2

10 9 8 8 7 6 6 5 5 3

10 9 8 7 7 6 6 5 4 3

Calculating Parameters and Statistics

Suppose ten students made the following scores on a five point quiz.
Calculate the measures of central tendency and variation.

Score

x

3

1

-2

4

2

2

-1

1

n = 10

1

2

-1

1

Mode = 3

3

3

0

0

Median = 3

5

3

0

0

= 3

4

3

0

0

Range = 4

3

3

0

0

s2= 1.33

2

4

1

1

s = 1.155

4

4

1

1

3

5

2

4

30

0

12

*Note: n - 1 used for small samples (usually less than 30 numbers).

Measures of Central Tendency

Mode = Number which occurs most often

Median = Middle number - 50% above - 50% below

Mean = Average = M = =

Measures of Variation

Range = Distance from highest to lowest score

Variance = Squared standard deviation = s2 =

Standard deviation = average distance of individual numbers from the
mean -

s =

Normal Curve

Percentages

Percentages are used to relate how much a part is of the whole and
established a basis for comparing information from groups of unequal sizes.

Example

If there is a 100 question test and we correctly answer 80 of them, our percentage of
correct answers is

Group A had 36 (50%) students who agreed with the statement while Group B had 25 (75%)
students who agreed.